It may be tough to come up with the numbers, but several rocket motors lost fuel, especially early on.

Kiwi-TNT was deliberately blown apart by a large reactivity transient, and Phoebus-1A had unplanned accident.

One of the earliest tests through out fuel rods and pieces of assembly. One of my colleagues has a video of that test. They used a sapphire mirror to look into the nozzle to see the core - and it was oscillating like a field of tall grass on a windy day.

Technically, if the core and fuel remained intact, then yes NERVA would be safe for a small payload if NERVA was a single stage, but it would probably be launched at a remote location.

As I recall, NERVA was planned as a second stage on something like a SATURN V booster.

The NERVA/ROVER programs were an outgrowth of the nuclear rocket propulsion for ICBMs. The original thermonuclear weapons systems were heavy, and the chemical rockets of the time could not deliver them 6-10K miles, so a nuclear rocket was considered the solution. During the development period in the 50's and into the 60's, the weapons designers successfully reduced the size of the thermonuclear warheads such that improved chemical rockets (e.g. Atlas) could deliver them the desired distance.

Thanks. Have they made any progress with miniaturization of the reactor (as well as correcting any issues the original had)? I'm wondering because if they did maybe we could make a single stage reusable spacecraft out of it, since I heard it could have a much greater thrust than traditional rockets.

The work on Nuclear Thermal Rockets (NTRs) pretty much ground to a halt in 1973. Much since then has been very limited with a lot of paper.

The Russians have had an ongoing project and folks from NASA and DOE have visited Russia as part of a collaborative effort. I'll see if I can dig up some information on what's been done.

A group (under profs Jim Tulenko and Samim Anghaie) down at The Innovative Nuclear Space Power and Propulsion Institute (INSPI) at the University of Florida, Dept. of Nuc. Eng. was doing some work on carbide fuels.
http://www.inspi.ufl.edu/ [Broken]

I guess what I was asking is assuming a successful firing, is a nerva rocket safe to use in an atmosphere?

Besides exhaust, when operated in an atmosphere, there's the problem of backscattered radiation (both gammas and neutrons) from the reactor reaching the payload/crew compartment. It's not just line of sight - radiation that escapes the reactor will scatter in any surrounding structure and medium and a portion will effectively go around and come in through the sides, and even the front, of the payload compartment. This isn't a problem in a vacuum though you'd still get scattering from the rocket structure.

To illustrate, here's an interesting paper on shielding requirements for nuclear submarines. It will help you see why the idea of the nuclear airplane didn't fly.